Abstract

The generation of internal-wave disturbances by continuously stratified flow of large depth over topography is studied in the Boussinesq approximation. For constant background velocity and Brunt-Vaisala frequency, the model of Long (1953) predicts a class of two-dimensional nonlinear steady-state wave patterns, under the hypothesis of no upstream effects from the topography. Here, Long's theory is generalized to describe the dynamics of slightly unsteady nonlinear disturbances near the hydrostatic limit (extended topography), allowing for the presence of weak variations in the background flow. A pair of nonlinear integral-differential amplitude-evolution equations are derived for disturbances of finite amplitude that remain valid as long as no flow reversal occurs. Previous weakly nonlinear or steady-state theories follow from this formulation as special cases. The proposed asymptotic theory seems well suited for investigating the long-time transient behaviour of the nonlinear response, and the validity of Long's hypothesis of no upstream influence.